6.2.  The Mass-Energy Equivalence Relation:

In addition to explaining the constancy and invariance of the speed of light and merging it with the second and third principles of Relativity, the Duality of Time postulate is the only way to explain the equivalence and transmutability between mass and energy (). Einstein gave various heuristic arguments for this relation without ever being able to prove it in any theoretical way (Hecht, 2011), neither did anyone else. Based on Doppler effect and Maxwell’s theory of radiation, the reasoning that he gave in his 1905 original derivation (Einstein, 1905) was questioned by Planck (Planck, 1906) and shown to be faulty (Ives, 1952). In 1907, Einstein acknowledged the controversy over his derivation (Einstein, 1907), and in the following forty years, he produced more than half dozen proofs that all suffer from unjustified assumptions or approximations. He never succeeded in producing a valid general proof (Ohanian, 2009). In 1955 he wrote in a letter to Carl Seelig: “I had already previously found that Maxwell’s theory did not account for the micro-structure of radiation and could therefore have no general validity.” (Capria, 2005). In 1990, Rohrlich tried to apply the relativistic Doppler shift, but he also had to introduce various approximations in order to reach the final equation (Rohrlich, 1990). Various other attempts have been made in this regard but until now there is no exact derivation of this famous formula.

It can be readily seen from Figure 6.2.1 that the transmutability between mass and energy can only occur in the inner levels of time, because it must involve motion at the speed of light that appears on the normal level of time as instantaneous, hence the same Relativity laws become inapplicable since they prohibit massive particles from moving at the speed of light, in which case , so the mass  would be infinite and also the energy. In the inner levels of time, however, this would be the normal behavior because the geometrical points are still massless, and their continuous coupling and decoupling is what generates mass and energy on the inner and outer levels of time, respectively, as explained further in section 6.2.3 below.

As we introduced in section 5.1 above, the normal limited velocities of massive physical particles and objects are a result of the spatial and temporal superposition of the various dual-state velocities of their individual points. This superposition occurs in the inner levels of time, where individually each point is massless and it is either at rest or moving at the speed of creation, but collectively they have some non-zero inertial mass , energy , and limited total apparent velocity , which can be calculated from equation 5. We also explained in section 5.3 above, that when we consider this imaginary velocity as being real, the Duality of Time Theory reduces to General Relativity, but when we consider its imaginary character we will uncover the hidden discrete space-time symmetry and we will automatically obtain Lorentz transformation, without introducing the principle of invariance of physics laws. For the same reason, we can see here that the mass-energy equivalence  can only be derived based on this profound discreteness that is manifested in dual-state velocity, which then allows the square integration in Figure 6.2.1, because the change in speed is occurring abruptly from zero to . Otherwise, when we consider  to be real continuous in time, we will get the gradual change which produces the triangular integration with the factor of half that gives the normal kinetic energy .

Based on this metaphysical behavior in the inner levels of time, we will provide in the following various exact derivations of the mass-energy equivalence relation, in its simple and relativistic forms, directly from the classical equation of mechanical work . The first two methods, in sections 6.2.2 and 6.2.3, involve integration (or rather: summation) in the inner time when the velocity changes abruptly from zero to , or when the mass is generated (from zero to ) in the inner time. This is obviously not allowed on the normal level of time when dealing with physical objects. The third method, in section 6.2.7, gives the total relativistic energy , by integrating over the inner and outer levels together, while in section 6.2.8 we will derive the relativistic energy-momentum relation directly from the definition of momentum as , also by integrating over the inner and outer levels together and accounting for what happens in each stage. Furthermore, we will see in sections 6.3 and 6.4 that the absolute invariance, and not just covariance, of complex momentum and energy, provide yet other direct derivations because they also lead to , that is equivalent with  or  as demonstrated in 9.

Actually, since we have shown previously that the new vacuum  is a perfect super-fluid, the mass-energy equivalence relation can be easily derived from the equation of wave propagation in such perfect medium: (), but we will not discuss that further in this article.